$2\cosec A$
Simplify Trigonometric Expression
The task is to simplify the following trigonometric expression:
\( \frac{\sin A}{1-\cos A} + \frac{1-\cos A}{\sin A} \)
Step-by-Step Simplification
-
Identify the common denominator for the two fractions, which is \((1-cos A)sin A\).
-
Rewrite the expression with the common denominator:
\( \frac{\sin^2 A}{(1-\cos A)\sin A} + \frac{(1-\cos A)^2}{\sin A (1-\cos A)} \) -
Combine the numerators over the common denominator:
\( \frac{\sin^2 A + (1-\cos A)^2}{(1-\cos A)\sin A} \) -
Expand the squared term in the numerator: $(1-cos A)2 = 1 - 2cos A + cos2 A$.
\( \frac{\sin^2 A + 1 - 2\cos A + \cos^2 A}{(1-\cos A)\sin A} \) -
Use the Pythagorean identity $sin2 A + cos2 A = 1$.
Substitute \(1\) for $sin2 A + cos2 A$:
\( \frac{1 + (1 - 2\cos A)}{(1-\cos A)\sin A} \) -
Simplify the numerator:
\( \frac{2 - 2\cos A}{(1-\cos A)\sin A} \) -
Factor out \(2\) from the numerator:
\( \frac{2(1 - \cos A)}{(1-\cos A)\sin A} \) -
Cancel the common factor \((1 - cos A)\), assuming \( A \neq 2n\pi \):
\( \frac{2}{\sin A} \) -
Apply the reciprocal identity \(cosec A = 1/sin A\):
\( 2\cosec A \)
Final Result
The simplified expression is \(2\cosec A\).
Similar Questions
What is the simplified value of the expression \(\frac{\cos A}{1-\tan A}+\frac{\sin A}{1-\cot A}\) ?
Which trigonometric ratio represents the opposite side divided by the hypotenuse in a right triangle?
As the angle increases beyond 45 degrees, what happens to the tangent value?
A ladder is leaning against a wall and makes an angle of 60° with the ground. If the length of the ladder is 24 m, find the distance of the foot of the ladder from the wall.
- $$ \frac{\sin 30^\circ}{1 + \cos 30^\circ} + \frac{1 + \cos 30^\circ}{\sin 30^\circ} = ? $$
- If ABCD is a Cyclic Quadrilateral, then the value of $\cos^2 A - \cos^2 B - \cos^2 C + \cos^2 D$ is:
- If $3\cos^2 x - 2\sin^2 x = -0.75$ and $0^\circ \le x \le 90^\circ$, then $x = ?$
Important Questions from Trigonometry
The value of 5 sin 14° sec 76° + 3 cot 15° cot 75° + 2 tan 45° is:
If two complimentary angles are in the ratio of 4 : 5, find the greater angle.
If \(\frac{\sin\spaceθ \space+\space \cos\spaceθ} {\sin \spaceθ \space-\space \cos \spaceθ} = \frac{\sqrt3 \space-\space 1}{\sqrt3 \space+\space 1} \) , then the angle θ is
If tan α = 1/2, tan β = 1/3, then find α + β.
Simplify: sin (A + B) sin (A – B)
